Question

Use the family in Problem 5 to find a solution of $\mathbf{xy}\mathbf{\text{'}\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{0}$that satisfies the boundary conditions$\mathbf{y}\left(0\right)\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(1\right)\mathbf{=}\mathbf{6}$.

Short answer

A solution satisfying the initial-value problem is $y=3x^2+1$.

Step-by-step solution

Define General solution of differential equation

The general solution of the differential equation$\mathrm{xy}\text{'}\text{'}-\mathrm{y}\text{'}=0$ is $\mathrm{y}={\mathrm{c}}_1+{\mathrm{c}}_2{\mathrm{x}}^2$, where are constants.

Find the value of the first constant

The initial conditions are:

$$\begin{gathered} \mathrm{y}\left(0\right)=1 \\ \mathrm{y}\text{'}\left(1\right)=6 \end{gathered}$$

Substituting the first condition in the general solution of the differential equation,

$$\begin{gathered} 1=c_1+c_2{\left(0\right)}^2 \\ {c}_1=1 \end{gathered}$$

Find the value of the second constant

Determining the first derivative of the general solution,

$y\text{'}=2c_{2}x$

Substituting the second condition in the first derivative of the general solution,

$$\begin{gathered} 6=2c_2\left(1\right) \\ 2c_2=6 \\ {c}_2=3 \end{gathered}$$

Find the value general solution satisfying the initial condition

Substituting the values of the constants in the general solution,

$$\begin{gathered} y=1+3x^2 \\ =3x^2+1 \end{gathered}$$

$\mathbf{Hence}\mathbf{}\mathbf{the}\mathbf{}\mathbf{solution}\mathbf{}\mathbf{is}\mathbf{}\mathbf{y}\mathbf{=}\mathbf{3}{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{1}$